Find The Domain Of The Expression:$\frac{x^2+6}{x^2-5x-6}$A. \[$ X \ \textless \ -1 \$\]B. \[$ X \neq 6, -1 \$\]C. \[$-1 \ \textless \ X \ \textless \ 6 \$\]D. All Real NumbersE. \[$ X \ \textgreater \ 6

Introduction

In mathematics, the domain of an expression is the set of all possible input values for which the expression is defined. When dealing with rational expressions, it's essential to consider the values that would make the denominator equal to zero, as division by zero is undefined. In this article, we will explore how to find the domain of the expression $\frac{x^2+6}{x^2-5x-6}$.

Understanding Rational Expressions

A rational expression is a fraction in which the numerator and denominator are polynomials. The domain of a rational expression consists of all real numbers except those that make the denominator equal to zero. To find the domain, we need to identify the values of $x$ that would make the denominator zero.

Finding the Values that Make the Denominator Zero

To find the values that make the denominator zero, we need to solve the equation $x^2-5x-6=0$. This is a quadratic equation, and we can solve it using the quadratic formula or factoring.

Factoring the Quadratic Equation

The quadratic equation $x^2-5x-6=0$ can be factored as $(x-6)(x+1)=0$. This tells us that either $x-6=0$ or $x+1=0$.

Solving for $x$

Solving for $x$, we get $x=6$ or $x=-1$.

Identifying the Domain

The domain of the expression $\frac{x^2+6}{x^2-5x-6}$ consists of all real numbers except $x=6$ and $x=-1$. Therefore, the domain is $x \neq 6, -1$.

Conclusion

In conclusion, finding the domain of a rational expression involves identifying the values that make the denominator equal to zero. By solving the quadratic equation $x^2-5x-6=0$, we found that the values $x=6$ and $x=-1$ make the denominator zero. Therefore, the domain of the expression $\frac{x^2+6}{x^2-5x-6}$ is $x \neq 6, -1$.

Answer

The correct answer is B. $x \neq 6, -1$.

Additional Examples

Here are a few more examples of finding the domain of rational expressions:

Example 1

Find the domain of the expression $\frac{x^2-4}{x^2+4x+4}$.

Solution

To find the domain, we need to solve the equation $x^2+4x+4=0$. This is a quadratic equation, and we can solve it using the quadratic formula or factoring.

Factoring the Quadratic Equation

The quadratic equation $x^2+4x+4=0$ can be factored as $(x+2)^2=0$. This tells us that $x+2=0$.

Solving for $x$

Solving for $x$, we get $x=-2$.

Identifying the Domain

The domain of the expression $\frac{x^2-4}{x^2+4x+4}$ consists of all real numbers except $x=-2$. Therefore, the domain is $x \neq -2$.

Example 2

Find the domain of the expression $\frac{x^2+2x-3}{x^2-9}$.

Solution

To find the domain, we need to solve the equation $x^2-9=0$. This is a quadratic equation, and we can solve it using the quadratic formula or factoring.

Factoring the Quadratic Equation

The quadratic equation $x^2-9=0$ can be factored as $(x-3)(x+3)=0$. This tells us that either $x-3=0$ or $x+3=0$.

Solving for $x$

Solving for $x$, we get $x=3$ or $x=-3$.

Identifying the Domain

The domain of the expression $\frac{x^2+2x-3}{x^2-9}$ consists of all real numbers except $x=3$ and $x=-3$. Therefore, the domain is $x \neq 3, -3$.

Example 3

Find the domain of the expression $\frac{x^2-4x+4}{x^2+2x+1}$.

Solution

To find the domain, we need to solve the equation $x^2+2x+1=0$. This is a quadratic equation, and we can solve it using the quadratic formula or factoring.

Factoring the Quadratic Equation

The quadratic equation $x^2+2x+1=0$ cannot be factored. We will use the quadratic formula to solve for $x$.

Solving for $x$

Using the quadratic formula, we get $x=\frac{-2 \pm \sqrt{2^2-4(1)(1)}}{2(1)}$. Simplifying, we get $x=\frac{-2 \pm \sqrt{0}}{2}$. This tells us that $x=\frac{-2}{2}=-1$.

Identifying the Domain

The domain of the expression $\frac{x^2-4x+4}{x^2+2x+1}$ consists of all real numbers except $x=-1$. Therefore, the domain is $x \neq -1$.

Final Answer

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Introduction

In our previous article, we discussed how to find the domain of a rational expression. We learned that the domain consists of all real numbers except those that make the denominator equal to zero. In this article, we will answer some frequently asked questions about finding the domain of rational expressions.

Q: What is the domain of a rational expression?

A: The domain of a rational expression is the set of all real numbers except those that make the denominator equal to zero.

Q: How do I find the domain of a rational expression?

A: To find the domain of a rational expression, you need to solve the equation that makes the denominator equal to zero. This is a quadratic equation, and you can solve it using the quadratic formula or factoring.

Q: What if the quadratic equation cannot be factored?

A: If the quadratic equation cannot be factored, you will need to use the quadratic formula to solve for $x$. The quadratic formula is $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What if the quadratic equation has no real solutions?

A: If the quadratic equation has no real solutions, then the denominator will never be equal to zero, and the domain of the rational expression will be all real numbers.

Q: Can I have a rational expression with a denominator that is always equal to zero?

A: No, you cannot have a rational expression with a denominator that is always equal to zero. This would mean that the expression is undefined for all values of $x$, which is not possible.

Q: How do I know if a rational expression is defined for a particular value of $x$?

A: To determine if a rational expression is defined for a particular value of $x$, you need to check if the denominator is equal to zero for that value of $x$. If the denominator is not equal to zero, then the expression is defined for that value of $x$.

Q: Can I have a rational expression with a denominator that is a perfect square?

A: Yes, you can have a rational expression with a denominator that is a perfect square. For example, the expression $\frac{x^2}{x^2+1}$ has a denominator that is a perfect square.

Q: How do I simplify a rational expression with a perfect square denominator?

A: To simplify a rational expression with a perfect square denominator, you can use the fact that the square root of a perfect square is equal to the absolute value of the number. For example, the expression $\frac{x^2}{x^2+1}$ can be simplified to $\frac{x^2}{(x+1)(x-1)}$.

Q: Can I have a rational expression with a denominator that is a difference of squares?

A: Yes, you can have a rational expression with a denominator that is a difference of squares. For example, the expression $\frac{x^2}{x^2-1}$ has a denominator that is a difference of squares.

Q: How do I simplify a rational expression with a difference of squares denominator?

A: To simplify a rational expression with a difference of squares denominator, you can use the fact that the difference of squares can be factored as $(x+1)(x-1)$. For example, the expression $\frac{x^2}{x^2-1}$ can be simplified to $\frac{x^2}{(x+1)(x-1)}$.

Q: Can I have a rational expression with a denominator that is a sum of squares?

A: No, you cannot have a rational expression with a denominator that is a sum of squares. This would mean that the expression is undefined for all values of $x$, which is not possible.

Q: How do I know if a rational expression is defined for a particular value of $x$?

A: To determine if a rational expression is defined for a particular value of $x$, you need to check if the denominator is equal to zero for that value of $x$. If the denominator is not equal to zero, then the expression is defined for that value of $x$.

Q: Can I have a rational expression with a denominator that is a product of two binomials?

A: Yes, you can have a rational expression with a denominator that is a product of two binomials. For example, the expression $\frac{x^2}{(x+1)(x-1)}$ has a denominator that is a product of two binomials.

Q: How do I simplify a rational expression with a product of two binomials denominator?

A: To simplify a rational expression with a product of two binomials denominator, you can use the fact that the product of two binomials can be factored as $(x+1)(x-1)$. For example, the expression $\frac{x^2}{(x+1)(x-1)}$ can be simplified to $\frac{x^2}{x^2-1}$.

Final Answer

The final answer is B. $x \neq 6, -1$.